I am working with Tamas Szamuely's book "Galois groups and Fundamental groups", and there is a point I am unsure in the proof of Lemma 5.6.2
The context is the following: $X$ is a quasi-compact and geometrically integral scheme (I think the second assumption might be irrelevant to the lemmma) over a field $k$. Let's fix $\overline{k}$ an algebraic closure of $k$ and $k_s$ a separable closure of $k$ sitting inside $\bar{k}$. Let $\overline{X}$ be the scheme $X \times_{\mathrm{Spec}(k)} \mathrm{Spec}(k_s)$.
Then lemma 5.6.2 states the following:
Given a finite étale cover $\overline{Y} \to \overline{X}$, there exists a finite subextension $L|k$ of $k_s$, and an étale cover $Y_L$ of $X_L := X_k \times_{\mathrm{Spec}(L)}\mathrm{Spec}(k_s)$, such that $\overline{Y} \cong Y_L \times_{\mathrm{Spec}(L)}\mathrm{Spec}(k_s)$
I interpret this lemma in the following way: since $k_s$ is the union of its finite subextensions, then $\mathrm{Spec}(k_s)$ can be viewed as the projective limit of the schemes $\mathrm{Spec}(L)$ where $L$ runs over the finite subextensions of $k_s$. The lemma then says that any finite étale cover of $\overline{X}$ actually comes from an étale cover of a subextension.
The proof goes this way: Since $X$ is quasi-compact, cover it by a finite number of affines $U_i = \mathrm{Spec}(A_i)$. Pullback these affines to $k_s$ to get a cover $\overline{U_i} = \mathrm{Spec}(A_i \otimes_k k_s)$ of $\overline{X}$. Now if $f : \overline{Y} \to \overline{X}$ is a finite etale morphism, then $f^{-1}(\overline{U}_i) = \mathrm{Spec}(B_i)$ where $B_i$ is some finite $A_i \otimes_{k} k_s$ algebra, so we can write $\mathrm{Spec}(B_i)$ as a quotient of $(A_i \otimes_k k_s)[x_1,\ldots,x_n]$.
Now the author claims that this quotient is generated by a finite number of polynomials. And that's where I am starting to have doubts. That would certainly be true if $A_i \otimes_k k_s$ were a noetherian ring, but with the given hypothesis, $A_i \otimes_k k_s$ is just an integral ring over $k_s$, so it has no reason to be noetherian, that would also be true if $B_i$ was of finite presentation over $k_s$.
Going on with that, the author takes a finite number of polynomials generating all of these algebras, and then take the subextenstion of $k_s$ generated by their coefficient. The author then claims the same can be done on the overlaps $U_i \cap U_j$ and that the isomorphisms $\overline{Y}_{U_i} \times_{X_i} \overline{Y}_{U_j} \cong \overline{Y}_{U_i \cap U_j}$ is defined by a finite number of equations, and that by taking an extension generated by the coefficients, we are done.
Once again, I have doubts, to me, it is not clear that the overlaps are defined by a finite number of equations.
I have looked in EGA, and EGA $IV_4$ 17.7.8 (plus EGA $IV_3$ 8.10.5 (x) for finiteness) states basically this lemma (that one can descend an etale morphism of a projective limit to an etale morphism at some index of the limit), but under the assumption that the morphisms of the components of the limit to the base schemes are of finite presentation. I have the feeling that this is a crucial hypothesis that have been forgotten in Szamuely's lemma, since in that case then indeed at least the ideals giving the $B_i$'s would be finitely generated.
So, first question: am I right in assuming that this hypothesis have been forgotten or is there some point that I missed?
Secondly: even in the case of finite presentation, the part concerning the overlaps of the $\overline{Y}_{U_i}$ is unclear to me. If the intersection $U_i \cap U_j$ is affine or just quasi-compact, then it is clear using the same method as above (writing it all in terms of a finite number of polynomial or covering by a finite number of things that can be written in terms of a finite number of polynomial, then taking the subextension generated by all the coefficients). But that would need some separation or quasi-separation hypothesis on $X$. The morphism $f$ is affine since it's finite, so it is separated, so that much might be useful when dealing with the inverse image of the intersection $\overline{U_i} \times_{\overline{X_i}} \overline{U_j}$.
But without some separation hypothesis, I do not see how to phrase the fact the the $\overline{Y}_{U_i}$ are compatible on the overlaps $U_i \cap U_j$ only using a finite number of coefficients, though it is surely possible since the proposition in EGA do not need such hypothesis.
Edit: I looked up in SGA I, there are no equivalent of lemma 5.6.2 in there. Szamuely's lemma 5.6.2 is used in the proof of the homotopy exact sequence for the etale fundamental group, which is SGA I IX 6.1. The hypotheses in SGA are the same as Szamuely's. So I might really be missing something here. The proof in SGA uses the fact that $\pi_1(\overline{X}, \overline{s}) = \varprojlim\pi_1(X_L, \overline{s_L})$, claiming this is essentially the fact that an étale cover of $\overline{X}$ comes from an étale cover of some $\overline{X}_L$ for sufficiently large $L$. Sadly for me, this fact is left to the reader in SGA I.
Edit 2: After some reflexion, I think that $f$ being locally of finite presentation (since it is etale) is enough to at least claim that each of the affines $\mathrm{Spec}(B_i)$ can be covered by a finite number spectras of $A_i \otimes_k k_s$-algebras of finite presentation, and so we can get away with "finitary" data as wanted. I still need to see if this work on the overlaps.
Edit 3: Thanks to KReiser, the fact that the ideals defining $B_i$ is finitely generated is solved. The compatibility on overlaps remains open.
On the confusing side, the homotopy exact sequence is stated in the Stacks Project (https://stacks.math.columbia.edu/tag/0BTX) with the additionnal assumption that $X$ is quasi-separated, which would solve the issue (see the comments in KReiser's answer), so there might be a forgotten hypothesis in Szamuely's book (but then also in SGA), and I don't know what to believe anymore.
Best Answer
First, just the fact that the morphism $f:\overline{Y}\to \overline{X}$ is etale and thus locally of finite presentation will give you the fact that $B_i\cong (A_i\otimes_k k_s)[x_1,\cdots,x_n]/(f_1,\cdots,f_m)$: by the standard methods, "locally of finite presentation" is local, so if $f:A\to B$ is a morphism of schemes which is locally of finite presentation and $\operatorname{Spec} S\subset A$ maps to $\operatorname{Spec} R\subset B$, then the induced map $R\to S$ is isomorphic to $R\mapsto R[x_1,\cdots,x_n]/(f_1,\cdots,f_m)$ for some finite $n,m$ (ref Stacks 01TQ part (2), for instance). Thus $B_i$ is of the desired form and there is no extra "covering" (as mentioned in your second edit) necessary.
For the situation involving the overlaps, I'm not sure about the correct resolution. The previous work on this matter in this post (which I've removed in this edit) was incorrect. Hopefully I will be able to return soon with an explanation.
After some time, I asked about the quasi-separatedness assumption on MathOverflow. Niels answered there:
It seems very reasonable to me that this assumption is necessary and was (accidentally) omitted in SGA1 and Szamuely's text.